We address in this paper Reinforcement Learning (RL) among agents that are grouped into teams such that there is cooperation within each team but general-sum (non-zero sum) competition across different teams. To develop an RL method that provably achieves a Nash equilibrium, we focus on a linear-quadratic structure. Moreover, to tackle the non-stationarity induced by multi-agent interactions in the finite population setting, we consider the case where the number of agents within each team is infinite, i.e., the mean-field setting. This results in a General-Sum LQ Mean-Field Type Game (GS-MFTGs). We characterize the Nash equilibrium (NE) of the GS-MFTG, under a standard invertibility condition. This MFTG NE is then shown to be $\mathcal{O}(1/M)$-NE for the finite population game where $M$ is a lower bound on the number of agents in each team. These structural results motivate an algorithm called Multi-player Receding-horizon Natural Policy Gradient (MRPG), where each team minimizes its cumulative cost independently in a receding-horizon manner. Despite the non-convexity of the problem, we establish that the resulting algorithm converges to a global NE through a novel problem decomposition into sub-problems using backward recursive discrete-time Hamilton-Jacobi-Isaacs (HJI) equations, in which independent natural policy gradient is shown to exhibit linear convergence under time-independent diagonal dominance. Experiments illuminate the merits of this approach in practice.

Self-play (SP) is a popular multi-agent reinforcement learning (MARL) framework for solving competitive games, where each agent optimizes policy by treating others as part of the environment. Despite the empirical successes, the theoretical properties of SP-based methods are limited to two-player zero-sum games. However, for mixed cooperative-competitive games where agents on the same team need to cooperate with each other, we can show a simple counter-example where SP-based methods cannot converge to a global Nash equilibrium (NE) with high probability. Alternatively, Policy-Space Response Oracles (PSRO) is an iterative framework for learning NE, where the best responses w.r.t. previous policies are learned in each iteration. PSRO can be directly extended to mixed cooperative-competitive settings by jointly learning team best responses with all convergence properties unchanged. However, PSRO requires repeatedly training joint policies from scratch till convergence, which makes it hard to scale to complex games. In this work, we develop a novel algorithm, Fictitious Cross-Play (FXP), which inherits the benefits from both frameworks. FXP simultaneously trains an SP-based main policy and a counter population of best response policies. The main policy is trained by fictitious self-play and cross-play against the counter population, while the counter policies are trained as the best responses to the main policy's past versions. We validate our method in matrix games and show that FXP converges to global NEs while SP methods fail. We also conduct experiments in a gridworld domain, where FXP achieves higher Elo ratings and lower exploitabilities than baselines, and a more challenging football game, where FXP defeats SOTA models with over 94% win rate.